Algebraic approach to Rump's results on relations between braces and pre-Lie algebras
aa r X i v : . [ m a t h . R A ] J u l Algebraic approach to Rump’s results on relationsbetween braces and pre-Lie algebras
Agata Smoktunowicz
Abstract
In 2014, Wolfgang Rump showed that there exists a correspondence betweenleft nilpotent right R -braces and pre-Lie algebras. This correspondence, estab-lished using a geometric approach related to flat affine manifolds and affine torsors,works locally. In this paper we explain Rump’s correspondence using only alge-braic formulae. An algebraic interpretation of the correspondence works for fieldsof sufficiently large prime characteristic as well as for fields of characteristic zero. In [17], Wolfgang Rump showed that there exists a correspondence between left nilpotentright R -braces and pre-Lie algebras. Braces were introduced by Rump in 2005 to de-scribe all involutive set-theoretic solutions of the Yang-Baxter equation. This approachsubsequently found applications in several other research areas.The algebraic formula for Rump’s connection from strongly nilpotent braces to pre-Lie algebras is particularly simple and enables rapid production of examples of pre-Liealgebras from braces. Question 1.
Is there a formula for a passage from finite pre-Lie algebras to fnite F -braces, where F is a finite field? The algebraic formula for a passage from pre-Lie algebras to F -braces in Rump’scorrespondence is not easy to use in examples, which raises the question: Question 2.
Is there an easy way to obtain braces from pre-Lie algebras?
In this paper all considered braces are left braces. Recall that if we change themultiplication in a left brace to the opposite multiplication we will obtain a right brace,and vice-versa. In Rump’s papers [16], [17], a brace always means a right brace. In [17]Rump presented a correspondence between right R -braces and RSA-the right symmetricalgebras. By taking the opposite multiplication in both the brace and the RSA it gives thecorrespondence between left-braces and LSA, left symmetric algebras, also called pre-Liealgebras.Notice that it is known that pre-Lie algebras are in correspondence with the ´etale affinerepresentations of nilpotent Lie algebras [6], and with Lie algebras with 1-cocycles [17],[4],[5], they also appear in noncommutative geometry. Braces have also been linked to otherresearch areas, for example, in [10], Gateva-Ivanova showed that there is a correspondencebetween braces and braided groups with an involutive braiding operator, whereas in [2],1achiller observed that there is a connection between braces and Hopf-Galois extensionsof abelian type (see also the appendix to [20] for some further results). In [17, 2, 7, 10, 20]braces and skew braces have been shown to be equivalent to several concepts in grouptheory (1-cocycles, regular subgroups, matched pairs of groups). Moreover, two-sidedbraces are exactly the Jacobson radical rings [16], [8]. In [9], applications of braces inquantum integrable systems were investigated, and in [18] R-matrices constructed frombraces were studied. Solutions of the reflection equation related to braces have also beeninvestigated by several authors.In this paper we use purely algebraic methods to look closely at Rump’s correspon-dence between braces and pre-Lie algebras and give algebraic formulas for this correspon-dence. A pre-Lie algebra A is a vector space with a binary operation ( x, y ) → xy satisfying( xy ) z − x ( yz ) = ( yx ) z − y ( xz ) , for every x, y, z ∈ A . We say that a pre-Lie algebra A is nilpotent if, for some n , allproducts of n elements in A are zero.Recall that a set B with binary operations + and ∗ is a left brace if ( A, +) is an abeliangroup and the following version of distributivity combined with associativity holds:( a + b + a ∗ b ) ∗ c = a ∗ c + b ∗ c + a ∗ ( b ∗ c ) , a ∗ ( b + c ) = a ∗ b + a ∗ c, moreover ( B, ◦ ) is a group where we define a ◦ b = a + b + a ∗ b .See [16] for the original definition. For a shorter equivalent definition using grouptheory see [8]. In what follows we will use the definition in terms of operation ’ ◦ ’ (presented in [8]): a set B with binary operations of addition +, and multiplication ◦ is abrace if ( A, +) is an abelian group, ( A, ◦ ) is a group and for every a, b, c ∈ Aa ◦ ( b + c ) + a = a ◦ b + a ◦ c. In [7], Catino and Rizzo defined F -braces thus: Let F be a field, let ( B, + , ◦ ) be a brace,then B is an F -brace if a ∗ ( eb ) = e ( a ∗ b ) for all a, b ∈ B, e ∈ F . Here a ∗ b = a ◦ b − a − b .In [16] Rump introduced left and right nilpotent braces and radical chains A i +1 = A ∗ A i and A ( i +1) = A ( i ) ∗ A , for a left brace A , where A = A = A (1) (the originalconstruction of Rump is for right braces, but we give the natural translation of it to leftbraces here). Recall that a left brace A is left nilpotent if there is a number n such that A n = 0, where inductively A i consists of sums of elements a ∗ b with a ∈ A, b ∈ A i − .A left brace A is right nilpotent if there is a number n such that A ( n ) = 0, where A ( i ) consists of sums of elements a ∗ b with a ∈ A ( i − , b ∈ A . Strongly nilpotent braces andthe chain of ideals A [ i ] of a brace A were defined in [19]. Define A [1] = A . A left brace A is strongly nilpotent if there is a number n such that A [ n ] = 0, where A [ i ] consists of sumsof elements a ∗ b with a ∈ A [ j ] , b ∈ A [ i − j ] for all 0 < j < i . A brace is strongly nilpotentif and only if it is both left nilpotent and right nilpotent [19].Various other radicals in braces were subsequently introduced, in analogy with ringtheory and group theory. Recall that solvable braces were introducted in [3], and in [14]2he connection between prime radical and solvable braces was investigated. See also [13]for some further results on solvable braces. In [12] the radical of a brace was introduced asthe intersection of all maximal ideals in a given brace. This radical enjoys good properties,and is very useful for describing the structure of a given brace. In [1], A.Agrachev and R. Gamkrelidze introduced the formal group of flows constructedfrom a pre-Lie algebra. Notice that this group of flows combined with the same additionis a brace, and it is the same brace as obtained in Rump’s correspondence. As mentionedby Rump in a private correspondence the addition in the pre-Lie algebra and in thecorresponding brace is always the same. In his survey [15], D. Manchon mentions thisgroup of flows along with explanations of their structure. To summarise from [1], [15],let A with operations · , + be a pre-Lie algebra over F , so a · ( b · c ) − ( a · b ) · c = b · ( a · c ) − ( b · a ) · c. Following the Rump correspondence [17], define the F -brace ( A, + , ◦ ) with the sameaddition as in pre-Lie algebra A and with the multiplication ◦ defined as in the group offlows as follows. The following is based on [1], [15], [17]. We additionally assume that A isa nilpotent pre-Lie algebra, and that F is a field of characteristic zero (or of characteristiclarger than the nilpotency index of A ). We use notation from [15].1. Let a ∈ A , and let L a : A → A denote the left multiplication by a , so L a ( b ) = a · b .Define L c · L b ( a ) = L c ( L b ( a )) = c · ( b · a ). Define e L a ( b ) = b + a · b + 12! a · ( a · b ) + 13! a · ( a · ( a · b )) + · · ·
2. We can formally consider element 1 such that 1 · a = a · a in our pre-Lie algebra(as in [15]) and define W ( a ) = e L a (1) − a + 12! a · a + 13! a · ( a · a ) + · · · Notice that W ( a ) : A → A is a bijective function, provided that A is a nilpotentpre-Lie algebra.3. Let Ω( a ) : A → A be the inverse function to the function W ( a ), so Ω( W ( a )) = W (Ω( a )) = a . Following [15] the first terms of Ω areΩ( a ) = a − a · a + 14 ( a · a ) · a + 112 a · ( a · a ) + . . .
4. Define a ◦ b = a + e L Ω( a ) ( b ) . Here, the addition is the same as in the pre-Lie algebra A . It was shown in [1] that( A, ◦ ) is a group. It is immediate to see that ( A, ◦ , +) is a left brace because a ◦ ( b + c ) + a = a + e L Ω( a ) ( b + c ) + a = ( a + e L Ω( a ) ( b )) + ( a + e L Ω( a ) ( c )) = a ◦ b + a ◦ c. Notice that the above correspondence works globally provided that A is a nilpotentpre-Lie algebra, so A n = 0 for some n .3hen the underlying pre-Lie algebra is a ring the obtained brace is with the familiarmultiplication a ◦ b = a + b + ab (see [15]). Remark about connections with the BCH formula.
Notice that the above formulacan also be written using the Baker-Campbell-Hausdorff formula and Lazard’s correspon-dence, see [1], [15] for details. In particular, the following formula for the multiplication ◦ holds in the brace obtained above: W ( a ) ◦ W ( b ) = W ( C ( a, b )) , where C ( a, b ) is obtained using the Campbel-Hausdorf series in the Lie algebra L ( A ).Recall that the Lie algebra L ( A ) is obtained from a pre-Lie algebra A by taking [ a, b ] = a · b − b · a , and has the same addition as A . By the Baker-Campbell-Hausdorff formula theelement C ( a, b ) can be represented in the form of a series in variables a, b , multiplicationby scalars and commutation in the Lie algebra L ( A ), and C ( a, b ) = a + b + + 12 [ a, b ] + 112 ([ a, [ a, b ]] + [ b, [ b, a ]]) + · · · . For more details see [1], page 1658, [15], page 3.
Example.
Let ( A, + , · ) be a pre-Lie algebra such that A [4] = 0. We calculate theformula for the multiplication in the corresponding brace ( A, + , ◦ ).We know that Ω( a ) = a − a · a + c for some c ∈ A [3] following the formula from [15].We obtain: e L Ω( a ) ( b ) = b + Ω( a ) · b + 12 Ω( a )(Ω( a ) · b ) + c ′ , where c ′ ∈ A [4] , so c ′ = 0. Therefore a ◦ b = a + b + a · b −
12 ( a · a ) · b + 12 a · ( a · b )hence a ∗ b = a · b −
12 ( a · a ) · b + 12 a · ( a · b )Observe that the following result follows from the above construction. Theorem 1.
Let ( A, + , · ) be a nilpotent pre-Lie algebra over a field F of characteristiczero, and let ( A, + , ◦ ) be the obtained brace as above. Denote a ∗ b = a ◦ b − a − b . Then a ∗ b = a · b + X x ∈ B α x x where α x ∈ F and B is the set of all products of elements a and b from ( A, · ) with b appearing only at the end, and a appearing at least two times in each product.Proof. This follows immediately from the construction of Ω( a ), which is a sum of a and a linear combination of all possible products of more than one element a with anydistribution of brackets, which can be proved by induction. Question 3.
What braces are obtained from the known types of pre-Lie algebras?
Question 4.
What braces are obtained from Novikov algebras? Some supporting lemmas
We recall Lemma 15 from [19]:
Lemma 2.
Let s be a natural number and let ( A, + , ◦ ) be a left brace such that A s = 0 forsome s . Let a, b ∈ A , and as usual define a ∗ b = a ◦ b − a − b . Define inductively elements d i = d i ( a, b ) , d ′ i = d ′ i ( a, b ) as follows: d = a , d ′ = b , and for i ≤ define d i +1 = d i + d ′ i and d ′ i +1 = d i d ′ i . Then for every c ∈ A we have ( a + b ) ∗ c = a ∗ c + b ∗ c + s X i =0 ( − i +1 (( d i ∗ d ′ i ) ∗ c − d i ∗ ( d ′ i ∗ c )) . Notation 1.
Let A be a strongly nilpotent brace with operations + , ◦ . Let x, y ∈ A .Consider elements x ∗ y , x ∗ ( x ∗ y ), . . . which are all products (with any distribution ofbrackets) of some non-zero number of elements x and one element y at the end. The setof all such elements will be denoted as E x,y . Notice that this set is finite, because A isa strongly nilpotent brace. We can list elements from the set E x,y in a such way thatshorter products always appear before longer products, and then we can make it into avector, which we will denote as V x,y .We will now prove a supporting lemma which will be useful in Section 5. In whatfollows, by 2 n c we denote the sum of 2 n copies of element c ∈ A . Lemma 3.
Let ( A, + , ◦ ) be a strongly nilpotent brace over the field of rational numbers.Then for every a, b ∈ A the limit lim n →∞ n ( n a ) ∗ b exists.Moreover, there are square matrices P , T , not depending on a, b , such that n ( 12 n a ) ∗ b = E P T n P − V a,b , where T is a matrix in the Jordan block form with the first Jordan block of dimension equal to and all other Jordan blocks with eigenvalues smaller than . Moreover E = [1 , , , . . . , and entries of S, T are rational numbers and n V n a,b = P T n P − V a,b . Proof.
Let notation for E x,y and V x,y be as in Notation 1 above. By Lemma 2 (appliedseveral times) every element from the set E x,y can be written as a linear combinationof elements from E x,y , with coefficients which do not depend on x and y . We can thenorganise these coefficients in a matrix, which we will call M = { m i,j } , so that we obtain M V x,y = V x,y . Notice that elements from E x,y (and from E x,y ) which are shorter appear beforeelements which are longer in our vectors V x,y and V x,y . Therefore by Lemma 2 it followsthat M is an upper diagonal matrix. Observe that the first element in the vector E x,y is (2 x ) ∗ y and that this element can be written as 2( x ∗ y ) plus elements of degree largerthan 2 (by Lemma 2). It follows that the first diagonal entry in M equals 2, so m , = 2.Observe that the following diagonal entries will be equal to 4 or more, because for example(2 x ) ∗ ((2 x ) ∗ y ) can be written using Lemma 2 as 4 x ∗ ( x ∗ y ) plus elements of degreelarger than 3. 5herefore M has exactly one eigenvalue equal to 2 with exactly one correspondingeigenvector, and all other eigenvalues equal 2 i for some i > M are its eigenvalues). Notice that M does not depend on x and y , as we only usedrelations from Lemma 2 to construct it. We can write M = P J P − where J consists ofJordan blocks, and the first block is the 1 × J , = 2. Notice that sincethe eigenvalues are real it is possible to find such matrices P, J with entries from the fieldof rational numbers.It follows that for every n , M n V x,y = V n x,y , therefore2 n V − n x,y = (2 M − ) n V x,y . Notice that 2 M − = P J ′ P − where J ′ = 2 J − is the matrix with Jordan blocks, and thefirst block is 1 × − i for i >
0. It follows that we can define the limit lim n →∞ n V − n x,y = lim n →∞ P J ′ n P − V x,y = P E , P − , where E , is the matrix with the first entry equal to 1 and all other entries equal to zero.The first entry of vector V − n x,y is (2 − n x ) ∗ y , therefore lim n →∞ n (2 − n x ) ∗ y = p , R V x,y ,where p , is the first diagonal entry of matrix P , and R is the first row of matrix P − . Notation 2.
Let A be a brace with operations + , ◦ , ∗ defined as usual so x ◦ y = x + y + x ∗ y . For x, y, z ∈ A and let E ( x, y, z ) ⊆ A denote the set consisting of anyproduct of elements x and y and one element z at the end of each product under theoperation ∗ , in any order, with any distribution of brackets, each product consisting ofat least 2 elements from the set { x, y } , each product having x and y appear at leastonce, and having element z at the end. Notice that E ( x, y, z ) is finite provided that A is a strongly nilpotent brace. Let V x,y,z be a vector obtained from products of elements x, y, z arranged in a such way that shorter products of elements are situated before longerproducts. Lemma 4.
Let ( A, + , ◦ ) be a strongly nilpotent brace over the field of rational numbers Q . Let a, b ∈ A . Denote a · b = lim n →∞ n ( 12 n a ) ∗ b. Then, for α, γ ∈ Q and a, b, c ∈ A we have ( αa + γb ) · c = α ( a · c ) + γ ( b · c ) and a · ( αb + γc ) = α ( a · b ) + γ ( a · c ) . Proof.
By the definition of a left Q -brace we immediately get that ( a · ( αb + γc )) = α ( a · b ) + γ ( a · c ).It remains to show that for a, b, c ∈ A we have a · ( b + c ) = a · b + a · c because thebase field is Q . Indeed, then for p, q ∈ Z , q = 0 we have ( pq a ) · b = p ((( p ) a · ) b ) = pq a · b .Observe also that by Lemma 2 ( a + b ) · c = lim n →∞ n ( n a + n b ) ∗ c = lim n →∞ n ( n a ) ∗ c + 2 n ( n b ) ∗ c + 2 n C ( n ) where C ( n ) is a sum of some products of elements n a and n b c at the end (because A is a strongly nilpotent brace). Moreover, eachproduct has degree at least 3.We need to show that lim n →∞ n C ( n ) = 0. We may consider a vector V − n a, − n b,c obtained as in Notation 2 from products of elements 2 − n a , 2 − n b, c .Using similar methods as in the proof of Lemma 3 we can show that for an appropriateupper-triangular matrix M with diagonal entries smaller than (equal to 2 − i for some i >
1) we have V n a, n b,c = M n V a,b,c , hence the limit 2 n V − n a, − n b,c exists and is equalzero, which implies that lim n →∞ n C ( n ) = 0. We now explain how to obtain a pre-Lie algebra from a brace. It is the same pre-Liealgebra as in Rump’s correspondence, but we developed an algebraic method to obtainthis algebra instead of geometric methods used by Rump. In a brace ( A, + , ◦ ) we willdenote as usual a ∗ b = a ◦ b − a − b . Theorem 5.
Let ( A, + , ◦ ) be a strongly nilpotent brace over the field of rational numbers.For a, b ∈ A define a · b = lim n →∞ n ( 12 n a ) ∗ b. Then ( A, + , · ) is a pre-Lie algebra.Proof. Observe first that that lim n →∞ n ( n a ) ∗ b exists by Lemma 3.We will now show that for every a, b, c ∈ A we have a · ( b · c ) − ( a · b ) · c = b · ( a · c ) − ( b · a ) · c. By Lemma 2 we get( x + y ) ∗ z = x ∗ z + y ∗ z + x ∗ ( y ∗ z ) − ( x ∗ y ) ∗ z + d ( x, y, z ) , ( y + x ) ∗ z = x ∗ z + y ∗ z + y ∗ ( x ∗ z ) − ( y ∗ z ) ∗ z + d ( y, x, z ) , where d ( x, y, z ) = E T V x,y,z for some vector E which does not depend of x, y, z , and where V x,y,z is as in Notation 2 (moreover d ( x, y, z ) is a combination of elements with at least 3occurences of elements from the set { x, y } ). It follows that x ∗ ( y ∗ z ) − ( x ∗ y ) ∗ z − y ∗ ( x ∗ z ) + ( y ∗ x ) ∗ z = d ( y, x, z ) − d ( x, y, z ) . Let a, b, c ∈ A and let m, n be natural numbers. Applying it to x = n a , y = m b , z = c we get ( 12 n a ) ∗ (( 12 m b ) ∗ c ) − (( 12 n a ) ∗ ( 12 m b )) ∗ c + d ( 12 n a, m b, c ) == ( 12 m b ) ∗ (( 12 n a ) ∗ c ) − (( 12 m b ) ∗ ( 12 n a )) ∗ c + d ( 12 m b, n a, c ) . By using the formula from Lemma 3 we obtain7 im m →∞ lim n →∞ m + n [( 12 n a ) ∗ (( 12 m b ) ∗ c ) − (( 12 n a ) ∗ ( 12 m b )) ∗ c + d ( 12 n a, m b, c )] == a · ( b · c ) − ( a · b ) · c and lim m →∞ lim n →∞ m + n [( 12 m b ) ∗ (( 12 n a ) ∗ c ) − (( 12 m b ) ∗ ( 12 n a )) ∗ c + d ( 12 m , n a, c )] = b · ( a · c ) − ( b · a ) · c. Consequently a · ( b · c ) − ( a · b ) · c = b · ( a · c ) − ( b · a ) · c. It remains to show that for α ∈ Q we have ( αa ) · b = α ( a · b ) and a · ( αb ) = α ( a · b ).It follows from Lemma 4. In this chapter we show that the correspondence between strongly nilpotent F -bracesand pre-Lie algebras over F is one-to-one for F = Q . Recall that N denotes the set ofnatural numbers. We start with the following. Proposition 6.
Let ( A, + , · ) be a nilpotent pre-Lie algebra over a field F of characteristiczero, and let ( A, + , ◦ ) be the brace obtained as in Section 3, so ( A, ◦ ) is the formal groupof flows of the pre-Lie algebra A . Suppose that F = R the field of real numbers (or thefield of rational numbers). Then for every a ∈ A there exists limit lim n →∞ n ( 12 n a ) ∗ b. Moreover a · b = lim n →∞ n ( 12 n a ) ∗ b, where n ∈ N .Proof. It follows immediately from the fact that the multiplication in a pre-Lie algebrais bilinear, and from the fact that a ∗ b can be expressed as in Theorem 1.We now obtain the ‘reverse’ theorem to Theorem 1: Theorem 7.
Let ( A, + , ◦ ) be a brace and let ( A, + , · ) be a nilpotent pre-Lie algebra overthe field Q obtained from this brace using Theorem 5, so a · b = lim n →∞ n ( n a ) ∗ b. Then ( A, ◦ ) is the group of flows of the pre-Lie algebra A , and ( A, + , ◦ ) can be obtainedas in Section 3 from pre-Lie algebra ( A, + , · ) . roof. Let E a,b be as in Notation 1. Observe that by Lemma 3 applied several times a · b = a ∗ b + X w ∈ E a,b α w w where α w ∈ F do not depend on w and each w is a product of at least 3 elements fromthe set { a, b } . Observe that coefficients α w do not depend on the brace A , as they wereconstructed using the formula from Lemma 2 which holds in every strongly nilpotentbrace (as we can consider V a,b to be an infinite vector with almost all entries zero in A ).Therefore a ∗ b = a · b − P w ∈ E a,b α w w , and now we can use this formula several times towrite every element from E a,b as a product of elements a and b under the operation · . Inthis way we can recover the brace ( A, + , ◦ ) from the pre-Lie algebra ( A, · , +).Notice that because we know that pre-Lie algebra ( A, + , · ) can be obtained as inTheorem 5 from the brace which is it’s group of flows (by Theorem 6) we can use thesame reasoning and ’recover’ the group of flows using the same formula.Therefore ( A, + , ◦ ) is the group of flows of pre-Lie algebra A .By combining results from Theorems 6 and 7 we get the following corollary. Corollary 8.
There is one-to-one correspondence between the set of strongly nilpotent Q -braces and the set of nilpotent pre-Lie algebras over Q . Acknowledgments.
I am very thankful to Wolfgang Rump for answering questionsabout his construction and for useful comments. This research was supported by theEPSRC grant EP/R034826/1.
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